= Mean accumulated hazard under independent censoring
{title2=$\mathbb E H(\min(T,C))=\mathbb P(T\leq C)$}
Let $X=\min(T,C)$, $V=\mathbf1_{\{T\leq C\}}$, and suppose $T$ and $C$ are independent, with continuous failure density $f=hS$. Then the <Tonelli theorem> gives
$$
\mathbb E H(X)=\int_0^\infty h(t)S(t)\mathbb P(C\geq t)dt=\mathbb P(T\leq C)=\mathbb EV.
$$
Thus $V-H(X)$ has mean zero. The identity also holds conditionally on covariates when censoring is independent conditionally on them. For an estimated hazard, the mean error is controlled by $\mathbb E|\widehat H(X)-H(X)|$, not by pointwise fit alone.
// Target: survival-analysis.bigb
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